3.3.76 \(\int \frac {(c e+d e x)^3}{(a+b \text {ArcSin}(c+d x))^{7/2}} \, dx\) [276]

Optimal. Leaf size=442 \[ -\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{5 b d (a+b \text {ArcSin}(c+d x))^{5/2}}-\frac {4 e^3 (c+d x)^2}{5 b^2 d (a+b \text {ArcSin}(c+d x))^{3/2}}+\frac {16 e^3 (c+d x)^4}{15 b^2 d (a+b \text {ArcSin}(c+d x))^{3/2}}-\frac {16 e^3 (c+d x) \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \text {ArcSin}(c+d x)}}+\frac {128 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \text {ArcSin}(c+d x)}}+\frac {32 e^3 \sqrt {2 \pi } \cos \left (\frac {4 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {16 e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{15 b^{7/2} d}-\frac {16 e^3 \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{15 b^{7/2} d}+\frac {32 e^3 \sqrt {2 \pi } S\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{15 b^{7/2} d} \]

[Out]

-4/5*e^3*(d*x+c)^2/b^2/d/(a+b*arcsin(d*x+c))^(3/2)+16/15*e^3*(d*x+c)^4/b^2/d/(a+b*arcsin(d*x+c))^(3/2)-16/15*e
^3*cos(2*a/b)*FresnelC(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/b^(7/2)/d-16/15*e^3*FresnelS(2*(
a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)*Pi^(1/2)/b^(7/2)/d+32/15*e^3*cos(4*a/b)*FresnelC(2*2^(1/
2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(7/2)/d+32/15*e^3*FresnelS(2*2^(1/2)/Pi^(1/2
)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(4*a/b)*2^(1/2)*Pi^(1/2)/b^(7/2)/d-2/5*e^3*(d*x+c)^3*(1-(d*x+c)^2)^(1/
2)/b/d/(a+b*arcsin(d*x+c))^(5/2)-16/5*e^3*(d*x+c)*(1-(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsin(d*x+c))^(1/2)+128/15*e
^3*(d*x+c)^3*(1-(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsin(d*x+c))^(1/2)

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Rubi [A]
time = 0.72, antiderivative size = 442, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4889, 12, 4729, 4807, 4727, 3387, 3386, 3432, 3385, 3433} \begin {gather*} \frac {32 \sqrt {2 \pi } e^3 \cos \left (\frac {4 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {16 \sqrt {\pi } e^3 \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {\pi } \sqrt {b}}\right )}{15 b^{7/2} d}-\frac {16 \sqrt {\pi } e^3 \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{15 b^{7/2} d}+\frac {32 \sqrt {2 \pi } e^3 \sin \left (\frac {4 a}{b}\right ) S\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {128 e^3 \sqrt {1-(c+d x)^2} (c+d x)^3}{15 b^3 d \sqrt {a+b \text {ArcSin}(c+d x)}}-\frac {16 e^3 \sqrt {1-(c+d x)^2} (c+d x)}{5 b^3 d \sqrt {a+b \text {ArcSin}(c+d x)}}+\frac {16 e^3 (c+d x)^4}{15 b^2 d (a+b \text {ArcSin}(c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{5 b^2 d (a+b \text {ArcSin}(c+d x))^{3/2}}-\frac {2 e^3 \sqrt {1-(c+d x)^2} (c+d x)^3}{5 b d (a+b \text {ArcSin}(c+d x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^3/(a + b*ArcSin[c + d*x])^(7/2),x]

[Out]

(-2*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(5*b*d*(a + b*ArcSin[c + d*x])^(5/2)) - (4*e^3*(c + d*x)^2)/(5*b^2*
d*(a + b*ArcSin[c + d*x])^(3/2)) + (16*e^3*(c + d*x)^4)/(15*b^2*d*(a + b*ArcSin[c + d*x])^(3/2)) - (16*e^3*(c
+ d*x)*Sqrt[1 - (c + d*x)^2])/(5*b^3*d*Sqrt[a + b*ArcSin[c + d*x]]) + (128*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^
2])/(15*b^3*d*Sqrt[a + b*ArcSin[c + d*x]]) + (32*e^3*Sqrt[2*Pi]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b
*ArcSin[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d) - (16*e^3*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c +
d*x]])/(Sqrt[b]*Sqrt[Pi])])/(15*b^(7/2)*d) - (16*e^3*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b
]*Sqrt[Pi])]*Sin[(2*a)/b])/(15*b^(7/2)*d) + (32*e^3*Sqrt[2*Pi]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*
x]])/Sqrt[b]]*Sin[(4*a)/b])/(15*b^(7/2)*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[f*(x^2/d)],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[f*(x^2/d)], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3387

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]

Rule 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 4727

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 - c^2*x^2]*((a + b*ArcSin
[c*x])^(n + 1)/(b*c*(n + 1))), x] - Dist[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), Sin[
-a/b + x/b]^(m - 1)*(m - (m + 1)*Sin[-a/b + x/b]^2), x], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c}, x]
&& IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 4729

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 - c^2*x^2]*((a + b*ArcSin
[c*x])^(n + 1)/(b*c*(n + 1))), x] + (Dist[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcSin[c*x])^(n + 1)/
Sqrt[1 - c^2*x^2]), x], x] - Dist[m/(b*c*(n + 1)), Int[x^(m - 1)*((a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2
]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 4807

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] - Dist[f*(m/(b
*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], Int[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n + 1), x], x] /;
FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]

Rule 4889

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rubi steps

\begin {align*} \int \frac {(c e+d e x)^3}{\left (a+b \sin ^{-1}(c+d x)\right )^{7/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {e^3 x^3}{\left (a+b \sin ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \text {Subst}\left (\int \frac {x^3}{\left (a+b \sin ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}+\frac {\left (6 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d}-\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d}\\ &=-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {x}{\left (a+b \sin ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{5 b^2 d}-\frac {\left (64 e^3\right ) \text {Subst}\left (\int \frac {x^3}{\left (a+b \sin ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{15 b^2 d}\\ &=-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x) \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {128 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {\left (16 e^3\right ) \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{5 b^3 d}-\frac {\left (128 e^3\right ) \text {Subst}\left (\int \left (\frac {\cos (2 x)}{2 \sqrt {a+b x}}-\frac {\cos (4 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d}\\ &=-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x) \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {128 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}-\frac {\left (64 e^3\right ) \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (64 e^3\right ) \text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (16 e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{5 b^3 d}+\frac {\left (16 e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{5 b^3 d}\\ &=-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x) \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {128 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {\left (32 e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{5 b^4 d}-\frac {\left (64 e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (64 e^3 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {4 a}{b}+4 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (32 e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{5 b^4 d}-\frac {\left (64 e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (64 e^3 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {4 a}{b}+4 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d}\\ &=-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x) \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {128 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {16 e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{5 b^{7/2} d}+\frac {16 e^3 \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{5 b^{7/2} d}-\frac {\left (128 e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{15 b^4 d}+\frac {\left (128 e^3 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{15 b^4 d}-\frac {\left (128 e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{15 b^4 d}+\frac {\left (128 e^3 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{15 b^4 d}\\ &=-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x) \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {128 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {32 e^3 \sqrt {2 \pi } \cos \left (\frac {4 a}{b}\right ) C\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {16 e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{15 b^{7/2} d}-\frac {16 e^3 \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{15 b^{7/2} d}+\frac {32 e^3 \sqrt {2 \pi } S\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{15 b^{7/2} d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.57, size = 445, normalized size = 1.01 \begin {gather*} \frac {e^3 \left (-4 (a+b \text {ArcSin}(c+d x)) \left (e^{2 i \text {ArcSin}(c+d x)} (4 i a+b+4 i b \text {ArcSin}(c+d x))+4 \sqrt {2} b e^{-\frac {2 i a}{b}} \left (-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )^{3/2} \text {Gamma}\left (\frac {1}{2},-\frac {2 i (a+b \text {ArcSin}(c+d x))}{b}\right )+e^{-2 i \text {ArcSin}(c+d x)} \left (-4 i a+b-4 i b \text {ArcSin}(c+d x)+4 \sqrt {2} b e^{\frac {2 i (a+b \text {ArcSin}(c+d x))}{b}} \left (\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )^{3/2} \text {Gamma}\left (\frac {1}{2},\frac {2 i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )\right )+4 (a+b \text {ArcSin}(c+d x)) \left (e^{-4 i \text {ArcSin}(c+d x)} (-8 i a+b-8 i b \text {ArcSin}(c+d x))+e^{4 i \text {ArcSin}(c+d x)} (8 i a+b+8 i b \text {ArcSin}(c+d x))+16 b e^{-\frac {4 i a}{b}} \left (-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )^{3/2} \text {Gamma}\left (\frac {1}{2},-\frac {4 i (a+b \text {ArcSin}(c+d x))}{b}\right )+16 b e^{\frac {4 i a}{b}} \left (\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )^{3/2} \text {Gamma}\left (\frac {1}{2},\frac {4 i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )-6 b^2 \sin (2 \text {ArcSin}(c+d x))+3 b^2 \sin (4 \text {ArcSin}(c+d x))\right )}{60 b^3 d (a+b \text {ArcSin}(c+d x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^3/(a + b*ArcSin[c + d*x])^(7/2),x]

[Out]

(e^3*(-4*(a + b*ArcSin[c + d*x])*(E^((2*I)*ArcSin[c + d*x])*((4*I)*a + b + (4*I)*b*ArcSin[c + d*x]) + (4*Sqrt[
2]*b*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b])/E^(((2*I)*a)/b)
+ ((-4*I)*a + b - (4*I)*b*ArcSin[c + d*x] + 4*Sqrt[2]*b*E^(((2*I)*(a + b*ArcSin[c + d*x]))/b)*((I*(a + b*ArcSi
n[c + d*x]))/b)^(3/2)*Gamma[1/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b])/E^((2*I)*ArcSin[c + d*x])) + 4*(a + b*Arc
Sin[c + d*x])*(((-8*I)*a + b - (8*I)*b*ArcSin[c + d*x])/E^((4*I)*ArcSin[c + d*x]) + E^((4*I)*ArcSin[c + d*x])*
((8*I)*a + b + (8*I)*b*ArcSin[c + d*x]) + (16*b*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-4*I)*(a
+ b*ArcSin[c + d*x]))/b])/E^(((4*I)*a)/b) + 16*b*E^(((4*I)*a)/b)*((I*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[
1/2, ((4*I)*(a + b*ArcSin[c + d*x]))/b]) - 6*b^2*Sin[2*ArcSin[c + d*x]] + 3*b^2*Sin[4*ArcSin[c + d*x]]))/(60*b
^3*d*(a + b*ArcSin[c + d*x])^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1264\) vs. \(2(368)=736\).
time = 0.53, size = 1265, normalized size = 2.86

method	result	size
default	\(\text {Expression too large to display}\)	\(1265\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/60/d*e^3/b^3*(32*arcsin(d*x+c)^2*(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1
/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*Pi^(1/2)*b^2-32*arcsin(d*x+c)^2*(a+b*arcsin(d*x+c))^(1/2
)*sin(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(-2/b)^(1/2)*2^(1/2)*Pi^(1/
2)*b^2-128*arcsin(d*x+c)^2*(a+b*arcsin(d*x+c))^(1/2)*cos(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*
arcsin(d*x+c))^(1/2)/b)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*b^2+128*arcsin(d*x+c)^2*(a+b*arcsin(d*x+c))^(1/2)*sin(4*
a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*b^2+6
4*arcsin(d*x+c)*(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+
b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*Pi^(1/2)*a*b-64*arcsin(d*x+c)*(a+b*arcsin(d*x+c))^(1/2)*sin(2*a/b)*FresnelS(
2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(-2/b)^(1/2)*2^(1/2)*Pi^(1/2)*a*b-256*arcsin(d*x+
c)*(a+b*arcsin(d*x+c))^(1/2)*cos(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*
2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*a*b+256*arcsin(d*x+c)*(a+b*arcsin(d*x+c))^(1/2)*sin(4*a/b)*FresnelS(2*2^(1/2)/Pi
^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*a*b+32*(-2/b)^(1/2)*(a+b*arcsin
(d*x+c))^(1/2)*cos(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*Pi^(1/
2)*a^2-32*(a+b*arcsin(d*x+c))^(1/2)*sin(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1
/2)/b)*(-2/b)^(1/2)*2^(1/2)*Pi^(1/2)*a^2-128*(a+b*arcsin(d*x+c))^(1/2)*cos(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/
(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*a^2+128*(a+b*arcsin(d*x+c))^(1/2)*sin(
4*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*a^2
+32*arcsin(d*x+c)^2*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^2-64*arcsin(d*x+c)^2*sin(-4*(a+b*arcsin(d*x+c))/b+4*
a/b)*b^2+64*arcsin(d*x+c)*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a*b+8*arcsin(d*x+c)*cos(-2*(a+b*arcsin(d*x+c))/b
+2*a/b)*b^2-128*arcsin(d*x+c)*sin(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*a*b-8*arcsin(d*x+c)*cos(-4*(a+b*arcsin(d*x+c
))/b+4*a/b)*b^2+32*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a^2-6*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^2+8*cos(-2*
(a+b*arcsin(d*x+c))/b+2*a/b)*a*b-64*sin(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*a^2+3*sin(-4*(a+b*arcsin(d*x+c))/b+4*a
/b)*b^2-8*cos(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*a*b)/(a+b*arcsin(d*x+c))^(5/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

integrate((d*x*e + c*e)^3/(b*arcsin(d*x + c) + a)^(7/2), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{3} \left (\int \frac {c^{3}}{a^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{3} x^{3}}{a^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {3 c d^{2} x^{2}}{a^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {3 c^{2} d x}{a^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**3/(a+b*asin(d*x+c))**(7/2),x)

[Out]

e**3*(Integral(c**3/(a**3*sqrt(a + b*asin(c + d*x)) + 3*a**2*b*sqrt(a + b*asin(c + d*x))*asin(c + d*x) + 3*a*b
**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**2 + b**3*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**3), x) + Integr
al(d**3*x**3/(a**3*sqrt(a + b*asin(c + d*x)) + 3*a**2*b*sqrt(a + b*asin(c + d*x))*asin(c + d*x) + 3*a*b**2*sqr
t(a + b*asin(c + d*x))*asin(c + d*x)**2 + b**3*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**3), x) + Integral(3*c*
d**2*x**2/(a**3*sqrt(a + b*asin(c + d*x)) + 3*a**2*b*sqrt(a + b*asin(c + d*x))*asin(c + d*x) + 3*a*b**2*sqrt(a
+ b*asin(c + d*x))*asin(c + d*x)**2 + b**3*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**3), x) + Integral(3*c**2*
d*x/(a**3*sqrt(a + b*asin(c + d*x)) + 3*a**2*b*sqrt(a + b*asin(c + d*x))*asin(c + d*x) + 3*a*b**2*sqrt(a + b*a
sin(c + d*x))*asin(c + d*x)**2 + b**3*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**3), x))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^3/(b*arcsin(d*x + c) + a)^(7/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^3/(a + b*asin(c + d*x))^(7/2),x)

[Out]

int((c*e + d*e*x)^3/(a + b*asin(c + d*x))^(7/2), x)

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